R Notebook
How do sites vary in their nutrient resource profiles?
As a starting point, I characterized sites by their resource availability to get a better sense of the variation in resource supply across sites, as well as correlation between the three potential limiting resources manipulated through fertilization.
## Number of sites selected## [1] 91
Calculating site stoichiometric ratios
To characterize sites by their relative resource supply and imbalance, we can calculate both the total amount of resources available in a given site, as well as the relative imbalance of these ratios, per Cardinale et al. (2009).
In this method, site resource concentrations of any \(n\) nutrients are standardized to their \(z\)-scores, then projected onto a vector (\(\vec{1}\)) that represents a perfect balance of resource stoichiometry.
In a two-resource example, see panel (a) in the figure below from Lewandoska et al. (2016)
This schematic shows some site resource concentrations in resources 1 and 2, denoted by the vector \(r\), projected upon a “perfectly balanced” vector denoted by \(a\). The length of this projection, \(a\), represents total resource supply in the system, while the orthogonal vector, \(b\) represents resource imbalance.
As an aside, I think there’s an error in the caption of this figure, where \(b\) is listed as a function of cosh(), which should produce an angle, \(\theta\). This measure of \(\theta\) is likely what is actually being used, as is the case in the original Cardinale et al 2009. I’ve listed the formula below:
\[\theta = \cos^{-1}\frac{(\vec{x} \cdot \vec{y})}{\left\lVert\vec{x} \right\rVert \left\lVert\vec{y}\right\rVert} = \cos^{-1}\frac{a}{r}\]
I haven’t seen this done before in other publications, but I believe we can also use this information to figure out what is limiting relative to this ideal stoichiometric ratio. The vector \(b\), which is the portion of our observed vector \(r\) that is not accounted for by the projection onto \(a\) can be termed the “rejection”, defined simply by \(b = r - a\)
Characterizing sites by resource availability and imbalance
Following this framework to characterize site resource abundance and imbalance, I’ve plotted the two key components of this relationship below.
As a refresher:
- \(a\) = Total resource supply
- \(\theta\) = Resource imbalance
What I see here is that sites tend to vary quite widely in their resource availability and the relative abundance of these resources. Some sites, such jena.de and lead.us have very high resource supply rates and comparitively low imbalance, while others, such as sgs.us and hnvr.us, are highly skewed in their resource supply.
What causes this resource imbalance
As an extension of this method to quantify resource availability and imbalance, we can provide some sense of what specific nutrients drive resource imbalance. Here, I’ve plotted the relative contribution of our 3 key nutrients (NPK) to resource imbalance, standardized relative to the observed resource vector.
Again, there appears considerable variation in patterns of abundance and deficiency in key nutrients across the different sites. I’ve embedded this figure interactively to allow for exploration.
Points are jittered to avoid overlap. Different positions on the Y axis correspond to different nutrients. Points towards the right end of the X axis depict high resource abundance for a given site, towards the left for resource deficiency.
Does resource abundance predict community productivity?
As a simple proof of concept, I’ve plotted the relationship below between community productivity before treatment vs. total nutrient supply. It appears that there is a positive correlation between total nutrient supply (length of the “a” vector) and and mean pre-treatment productivity, though it’s not a perfect relationship and may be more of a saturating curve.
How well do soil resources partition observational community variance?
What sites exhibit the greatest proportion of community variance explained by soil nutrient Z-scores?
Here I’ve run a db-RDA (may have some favorable properties relative to RDA, but functions similarly using different sorts of distance measures, rather than just euclidean) on each site, which should allow us to determine:
- What fraction of the overall variance in observational community composition we can attribute to variation in soil N, P, and K (and to test this fraction with ANOVA)
- What is the estimated relationship between community composition and soil characteristics (plot scores on N, P, and K axes)
Because I’ve subset our data to just those plots containing 30 plots in three blocks, these comparisons should be done with very similar datasets - similar to Jon Bakker’s approach for the temporal turnover manuscript.
Here we see that a number of sites have significant fractions of community variance that can be explained by soil variables in db-RDA, and there’s quite a range therein.
What sites show the greatest amount of variance attributable to soil resource variation?
Relating this variance fraction to CV’s of N, P, and K shows an interesting, although not unexpected, pattern where the amoung of variance explained by soil nutrient Z-scores was correlated with the CV of nitrogen.
## Anova Table (Type II tests)
##
## Response: varfrac
## Sum Sq Df F value Pr(>F)
## CV_n 0.026736 1 3.7019 0.0687 .
## CV_p 0.000740 1 0.1025 0.7522
## CV_k 0.001911 1 0.2645 0.6127
## CV_n:CV_p 0.008095 1 1.1209 0.3023
## CV_n:CV_k 0.001484 1 0.2054 0.6553
## CV_p:CV_k 0.005696 1 0.7887 0.3850
## CV_n:CV_p:CV_k 0.003889 1 0.5385 0.4716
## Residuals 0.144444 20
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Correlations between environmental scores
If we look at community scores, we also see some evidence for corrrelation among responses to N, P, and K availability.
